Induction and Recursion

Abstract

Almost all of part I of this book belongs to elementary number theory (ENT). This notion can be rigorously defined using tools of mathematical logic, but in order to do this one must first introduce a formal language of arithmetic and fix an adopted system of axioms (one or other version of Peano’s axioms). In order to avoid such irrelevant details, we restrict ourselves to some intuitive remarks. In ENT there are some initial statements and some axioms, which formalize our intuitive ideas of natural numbers (or integers), as well as certain methods for constructing new statements and methods of proofs. The basic tool for construction is recursion. In the simplest case assume that we want to define some property P(n) of a natural number n. Using the method of recursion we explain how one can decide whether P(n + 1) is true if it is already known whether P(1),..., P(n) are true or not. Say, the property “n is a prime” can be defined as follows: “1 is not a prime; 2 is a prime; n +1 ≥ 3 is a prime iff none of the primes among 1, 2,..., n divide n + 1”. Analogously the main tool in the proofs of ENT is induction. In order to prove by induction a statement of type “Vn, P(n) is true” we first prove say P(l) and then the implication “Vn the property P(n) implies P(n + 1)”.